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Dynamics of state-wise prospective reserves in

the presence of non-monotone information

Marcus C. Christiansen1 and Christian Furrer2,3,⋆

1Institut fur Mathematik, Carl von Ossietzky Universitat Oldenburg, Carl-von-Ossietzky-Straße 9–11, DE-26129

Oldenburg, Germany.2PFA Pension, Sundkrogsgade 4, DK-2100 Copenhagen Ø, Denmark.

3Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø,

Denmark.⋆Corresponding author. E-mail: furrer@math.ku.dk.

In the presence of monotone information, stochastic Thiele equations describing thedynamics of state-wise prospective reserves are closely related to the classic martin-gale representation theorem. When the information utilized by the insurer is non-monotone, classic martingale theory does not apply. By taking an infinitesimal ap-proach, we derive generalized stochastic Thiele equations that allow for informationdiscarding. The results and their implication in practice are illustrated via exampleswhere information is discarded upon and after stochastic retirement.

Keywords: Life insurance; Stochastic Thiele equations; Infinitesimal martingales; Marked pointprocesses; Stochastic retirement

JEL Classification: G22; C02

1. Introduction

Life insurers frequently employ reduced information in the valuation of liabilities due to e.g. legalconstraints and data privacy considerations or to achieve model simplifications. The possibilityof information discarding leads to potentially decreasing flows of information for which classicmartingale theory does not apply. Based on the novel infinitesimal approach proposed and deve-loped in Christiansen (2020), we study the dynamics of so-called state-wise prospective reservesin the presence of non-monotone information. Our main contribution is a generalization of thestochastic Thiele equations of Norberg (1992, 1996) to allow for non-monotone information. Sec-ondary contributions include a careful study of the concept of state-wise prospective reserves anda discussion of current actuarial practices regarding valuation in relation to information discardingupon and after stochastic retirement.

In this paper, the only source of randomness consists of the state of the insured, which is modeledas a non-explosive pure jump process on a finite state space. This places our work within thefield of multi-state life insurance mathematics. The definitions of retrospective and prospectivereserves in Norberg (1991) encompass non-monotone information, and under (semi-)Markovian

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assumptions specific instances of non-monotone information appear in the study of retrospectivereserves and bonus prognosis, see Norberg (1991, 1999, 2001) and Helwich (2008). But to ourknowledge, the literature contains no attempts at the development of a unifying theory for non-Markovian models under non-monotone information. Our contribution constitutes the first steptowards this goal, since we impose no restrictions on the intertemporal dependency structure andallow for general information discarding occurring at stopping times w.r.t. the state of the insured.

The multi-state approach to life insurance dates back at least to Hoem (1969), where Thieleequations describing the dynamics of the state-wise prospective reserves are derived under theassumption that the process governing the state of the insured is Markovian. These differen-tial/integral equations were revisited by Norberg in his seminal paper Norberg (1991) and havesince been generalized in various directions. This includes relaxing the assumption of Markovianityto allow for duration dependency (semi-Markovianity), taking market risks into account, and thestudy of higher order moments of prospective reserves, see e.g. Møller (1993), Steffensen (2000),Helwich (2008), Adekambi and Christiansen (2017), and Bladt et al. (2020). We should mentionthat while the approach of Steffensen (2000) is very general, the results are only established understrict smoothness conditions that might not be satisfied in practice.

The ordinary Thiele equations are essentially Feynman-Kac type results. In contrast, thestochastic Thiele equations of Norberg (1992, 1996) are stochastic differential equations thatapply irregardless of the intertemporal dependency structure and reveal the universality of Thiele’soriginal equation. Furthermore, under Markovian assumptions, stochastic Thiele equations canbe used to elegantly derive Feynman-Kac formulas for the prospective reserve.

In the presence of monotone information, the dynamics of prospective reserves are characterizedby identifying integrands in the classic martingale representation theorem for random countingmeasures (Norberg, 1992, 1996; Christiansen and Djehiche, 2019). In similar fashion, our ap-proach relies on the infinitesimal martingale representation theorem of Christiansen (2020), whichextends the classic martingale representation theorem for random counting measures to allow fornon-monotone information. Essentially, our methodology and results accompany Christiansen(2020); while Christiansen (2020) contains the general theory for so-called infinitesimal compen-sators and infinitesimal martingales, this theory is here applied to multi-state life insurance.

Although we focus on state-wise prospective reserves and their dynamics, we expect the settingand mathematical techniques presented here to be applicable beyond this specific application,e.g. in relation to estimation and efficient computation of expected cash flows and reserves in thepresence of non-monotone information. Broadly speaking, with this paper we initialize a programthat aims at the development of general mathematical methodology for multi-state life insurancein the presence of non-monotone information.

The paper is structured as follows. In Section 2, we present the probabilistic setup and themain examples concerning information discarding upon and after retirement. In Section 3, wedevelop a mathematically sound concept of state-wise prospective reserves in the presence ofpotentially non-monotone information. Section 4 contains our main result, namely a generalizationof the stochastic Thiele equations to allow for non-monotone information, and its application toinformation discarding upon and after retirement. In particular, we illustrate the pertinenceand usefulness of the generalized stochastic Thiele equations by deriving Feynman-Kac formulasbeyond the (semi-)Markovian case.

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2. Monotone and non-monotone information structures

In this section, we introduce a general modeling framework for the random pattern of states of theinsured in the presence of non-monotone information. The framework is strongly related to thegeneral theory of non-monotone information for jump processes introduced by Christiansen (2020).To clarify the theoretical as well as practical relevance of an approach allowing for non-monotoneinformation and general intertemporal dependency structures, we further discuss a motivatingexample concerning stochastic retirement. This leads to the specification of some explicit casesof non-monotone information that serve as the main examples in the ensuing investigation.

2.1. General setting

Let (Ω,A, P ) be a complete probability space with null sets N , and let Z = (Zt)t≥0 be a randompattern of states (pure jump process) on the finite state space S = 1, . . . , J + 1, J + 2, J ∈ N0,with initial state Z0 ≡ z0 ∈ S, giving at each time t the state of the insured in S.

The total information available is denoted F = (Ft)t≥0; it is the right-continuous and completefiltration given by

Ft = σ(Zs : s ≤ t) ∨ N .

Since F is a filtration, it represents monotone (increasing) information.We relate to the random pattern of states Z a multivariate counting process N = (N(t))t≥0

with components Njk = (Njk(t))t≥0, j, k ∈ S, j 6= k, giving the number of jumps of Z from statej to state k:

Njk(t) = # s ∈ (0, t] : Zs− = j, Zs = k, t ≥ 0.

We impose the following technical condition. It ensures that Z is non-explosive and that com-pensated counting processes are true martingales.

Assumption 2.1 (No explosions and true martingales). We assume that

E[

∑

j,k∈Sj 6=k

Njk(t)]

<∞

for all t ≥ 0. ⋄

If we denote by T (t) the next jump after time t,

T (t) = infs ∈ (t,∞) : Zs 6= Zt,

T (∞) = ∞,

and employ the convention inf ∅ = ∞, we can also define a marked point process (τi, Zτi)i∈N0by

τ0 = 0, τi = T (τi−1), i ∈ N,

with Z∞ = ∇ for some arbitrary cemetery state ∇. The marked point process, multivariatecounting process, and random pattern of states formulations of the setup are equivalent in thesense that the information generated by these processes agree.

A life insurance contract between the insured and the insurer is stipulated by the specificationof a payment process B = (B(t))t≥0 representing the accumulated benefits minus premiums. Ingeneral, we suppose that B is an F-adapted process that has cadlag sample paths, finite expectedvariation on compacts (in particular, it has sample paths of finite variation on compacts), and adeterministic initial value B(0) ∈ R.

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2.2. Non-monotone information

Due to e.g. legal constraints, privacy considerations, or to achieve model and/or computationalsimplifications, the insurer might not have access to or desire to utilize all information available toit. Examples include the newly introduced General Data Protection Regulation 2016/679 of theEuropean Union, where Article 17 describes a so-called right to erasure, and the restriction to aMarkovian type of information even when the Markov property is not satisfied. Representing theresulting utilized information as a subsequence of σ-algebras, one typically finds that the sequenceis non-monotone because certain pieces of information are discarded en route.

To describe the information reductions, we introduce a subsequence of σ-algebras as follows.Let (Ti)i∈N and (Si)i∈N be sequences of F-stopping times with Si ≥ Ti, i ∈ N. Further, let(ζi)i∈N be a sequence of random variables with values in a separable complete metric space E andcorresponding Borel σ-algebra E := B(E), and suppose that each ζi is FTi-measurable. For thesake of a convenient notation, without loss of generality we assume that 0 6∈ E. The informationζi is recorded at time Ti and then discarded at a later time Si; here Si = ∞ signifies no discarding.Thus the admissible information at time t ≥ 0 is given by the σ-algebra Gt ⊆ Ft defined by

Gt = σ(Ti ≤ t < Si ∩ ζi ∈ A : i ∈ N, A ∈ E) ∨N , (2.1)

while the information available immediately before time t > 0 is given by the σ-algebra Gt− ⊆ Ft−

defined by

Gt− = σ(Ti < t ≤ Si ∩ ζi ∈ A : i ∈ N, A ∈ E) ∨ N . (2.2)

We introduce the notation G = (Gt)t≥0 and G− = (Gt−)t>0.The subsequence of σ-algebras G = (Gt)t≥0 is in general non-monotone and the random times

Ti and Si are not necessarily stopping times w.r.t. G. We do not assume the random times (Ti)i∈Nand (Si)i∈N to take a specific order in time other than Ti ≤ Si, and we even allow for simultaneousevents. We can recover F by taking Si = ∞, Ti = τi, and ζi = (τi, ZTi) for all i ∈ N, and fromthis point and onward, that representation is always assumed whenever G = F .

Let S := x ⊂ N : |x| < ∞ be the finite subsets of the natural numbers. Note that S iscountable. For each x ∈ S we define the indicator processes

Ix(t) :=

1 :⋂

i∈xTi ≤ t < Si ∩⋂

i 6∈x(Ω \ Ti ≤ t < Si),

0 : else,

so that Ix(t) is Gt-measurable for each t ≥ 0 and x ∈ S. We assume in continuation of Assump-tion 2.1 that

E

[ ∞∑

i=1

1Ti≤t

]

<∞, t ≥ 0, (2.3)

which implies that on each compact interval we can almost surely find at most finitely manyrandom times Ti, Si, i ∈ N. As a result, the indicator processes Ix have cadlag paths of finitevariation on compacts. The family of indicator processes I := (Ix)x∈S corresponds to the G-adapted non-explosive random pattern of states

Zt =∑

x∈S

x Ix(t), t ≥ 0.

4

This random pattern of states describes the state of information: Zt = x if and only if exactlythe information (ζi)i∈x is available at time t; in particular, the information (ζi)i/∈x has either beenrecorded and already discarded or is yet to be recorded.

We generally suppose that

σ(Zt) ⊆ Gt, t ≥ 0. (2.4)

Since we assumed that 0 6∈ E, the information at time t and at time t− can be alternativelyrepresented as

Gt =σ(ζxIx(t) : x ∈ S) ∨ N , t ≥ 0,

Gt− =σ(ζxIx(t−) : x ∈ S) ∨ N , t > 0,(2.5)

where ζx := (ζi)i∈x, x ∈ S. Let

Txy := inft ≥ 0 : Ix(t−)Iy(t) = 1,

using the convention inf ∅ := ∞. We see that Txy is the exact point in time where the stateof information changes from state x to state y by discarding information ζx\y and recordinginformation ζy\x; here we ignore information that is recorded and immediately discarded. Thetotal information either discarded or recorded at time Txy is thus ζxy := (ζi)i∈x∆y, where x∆y =(x \ y) ∪ (y \ x).

The extended marked point process (Ti, Si, ζi)i∈N corresponds to the random counting measuresνxy, x, y ∈ S, y 6= x, defined as the unique completions of

νxy([0, t] ×A) := 1Txy≤t1ζxy∈A, t ≥ 0, A ∈ B(Exy),

where Exy := E|x∆y|.If Ti = τi for all i ∈ N, the σ-algebra Gt reveals in particular the indices i of the admissi-

ble observations and thus gives a lower bound on the number of past discards, cf. Remark 3.1in Christiansen (2020), which might be an unwanted feature. As further discussed in Christiansen(2020), by considering suitable permutations it is often possible to obtain non-informative indices;in that case, the number of past discards becomes non-admissible information. In the next sub-section, we introduce some specific instances of non-monotone information concerning stochasticretirement and embed them into the framework above. In particular, we exemplify how to obtainnon-informative indices using suitable permutations.

2.3. Stochastic retirement

Suppose J ≥ 1 and z0 /∈ J + 1, J + 2, and let δ and η be the first hitting times of J + 2 andJ + 1, respectively, by Z:

δ = inft ≥ 0 : Zt = J + 2,

η = inft ≥ 0 : Zt = J + 1.

We think of δ as the time of death and η as the time of retirement. Accordingly, the states1, . . . , J describe the health state of the insured up until retirement or death. In this subsection,we assume a decrement structure such that retirement occurs at most once and death is a terminalevent:

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Assumption 2.2 (Decrement structure concerning retirement and death). We assume that

[0,∞) ∋ t 7→∑

j∈Sj>J

∑

k∈Sk≤J

Njk(t) = 0,

[0,∞) ∋ t 7→ N(J+2)(J+1)(t) = 0,

almost surely. ⋄

Note that the structure of the state space entails that the insurer is not updating its informationconcerning the health state of the insured upon or after retirement. In Figure 2.1 we haveexemplified this setup for the case J = 2 corresponding to a disability model allowing for recoverybefore retirement.

disabled 2active 1

retired 3

dead 4

Figure 2.1: Extension of classic disability model with recovery to allow for stochastic retirement.

In actuarial practice, it is common to impose some Markovian structure by assuming the randompattern of states Z to be e.g. Markovian or semi-Markovian. In the following, we illustrate whysuch assumptions might be insufficient and, as an alternative, how to represent similar assump-tions as non-monotone information substructures. This motivates the general non-Markovianframework with non-monotone information introduced in Subsections 2.1–2.2.

It is natural to imagine the random pattern of states Z as embedded into a larger framework.Let Z be a random pattern of states on an extended state space S = 1, . . . , J+1, J+2, . . . , 2J+1with initial state Z0 = z0 ∈ 1, . . . , J. Denote the corresponding multivariate counting processby N . Suppose that

E[

∑

j,k∈Sy 6=x

Njk(t)]

<∞ (2.6)

for all t ≥ 0, and that

[0,∞) ∋ t 7→∑

j∈Sj>J

∑

k∈Sk≤J

Njk(t) = 0,

[0,∞) ∋ t 7→∑

k∈SJ<k≤2J

N(2J+1)k(t) = 0,

almost surely. We think of the states J + 1, . . . , 2J as providing information concerning thehealth state of the insured upon or after retirement. In Figure 2.2 we have exemplified this

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setup for the case J = 2 corresponding to a disability model allowing for recovery and stochasticretirement. In general, we can now redefine Z by

Zt =

Zt if Zt ∈ 1, . . . , J,

J + 1 if Zt ∈ J + 1, . . . , 2J,

J + 2 if Zt = 2J + 1

for all t ≥ 0, when we find that z0 /∈ J +1, J +2 and that Assumptions 2.1–2.2 remain satisfied.

disabled 2active 1

retired& disabled

4retired& active

3

dead 5

Figure 2.2: Extension of the disability model with retirement of Figure 2.1 where the health statusof the insured remains observed upon and after retirement.

The information available to the insured is represented by the filtration F = (Ft)t≥0 given by

Ft = σ(Zs : s ≤ t) ∨ N .

In many cases, the information F is not available to the insurer, and then the insurer must resortto the information given by F ; this can e.g. be the case if upon retirement, disability coverageceases.

It appears consistent with actuarial practice to propose that the underlying random pattern ofstates Z is Markovian or semi-Markovian. We now study the resulting implications on Z, whichis the natural modeling object given information F . Let U = (Ut)t≥0 and U = (Ut)t≥0 be theduration processes associated with Z and Z, respectively, given by

Ut = t− sups ∈ [0, t] : Zs 6= Zt,

Ut = t− sups ∈ [0, t] : Zs 6= Zt.

Note that 1t≤ηUt = 1t≤ηUt. Let Ur = (U r

t )t≥0 be the time since retirement given by

U rt =

0 if t < η,

t− η if t ≥ η,

let H = (Ht)t≥0 be the state of the insured just before retirement given by

Ht =

Zt if t < η,

Zη− if t ≥ η,

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and let Uh = (Uht )t≥0 be the duration of the latest sojourn before retirement given by

Uht =

Ut if t < η,

Uη− if t ≥ η.

Proposition 2.3. Suppose (Z, U) is Markovian. Then (Z,Uh, U r,H) is Markovian. Suppose

further that Z is Markovian. Then (Z,U r ,H) is Markovian.

Proof. See Appendix A.

It is possible to derive necessary and sufficient conditions for which (semi-)Markovianity ofZ implies (semi-)Markovianity of Z, see e.g. Serfozo (1971). In general, these conditions arevery restrictive and do not apply to models of actuarial relevance: in this sense, the complexintertemporal dependency structure implied by Proposition 2.3 must be taken into account. Thisserves as a motivation for the general non-Markovian framework presented in Subsection 2.1.

Although Proposition 2.3 indicates that the mortality as retiree might depend on the pastthrough e.g. the time since retirement and the last health state before retirement, it is commonin actuarial practice to rely on a standard mortality table – an example is the longevity bench-mark of the Danish financial supervisory authority, cf. Jarner and Møller (2015). This in a sensecorresponds to imposing an ‘as if’ Markovian assumption or, alternatively, to only utilize infor-mation corresponding to a specific subsequence of σ-algebras rather than F itself. Therefore, weintroduce two subsequences of σ-algebras G1 = (G1

t )t≥0 and G2 = (G2t )t≥0 given by

G1t = σ(Zs1Zt∈1,...,J,1η≤s,1δ≤s : s ≤ t) ∨N ,

G2t = σ(Zs1Zt∈1,...,J,1η≤t,1δ≤s : s ≤ t) ∨ N .

The information G1 corresponds to the case where upon retirement or death the insurer discardsthe previous health records of the insured. The sub-information G2 ⊂ G1 even keeps no recordon the time of retirement. For most if not all practical purposes, the discarding of previousinformation upon death is of no importance.

Further, for describing the admissible information immediately before time t ≥ 0 we definesequences of σ-algebras G1

− = (G1t−)t≥0 and G2

− = (G2t−)t≥0 by

G1t− = σ(Zs1Zt−∈1,...,J,1η<s,1δ<s : s ≤ t) ∨ N ,

G2t− = σ(Zs1Zt−∈1,...,J,1η<t,1δ<s : s ≤ t) ∨ N .

Lemma 2.4. The σ-algebras G1t , G

1t−, G

2t , and G2

t−, t ≥ 0, can be brought on the form of (2.1)–(2.2).

Proof. See Appendix A.

Lemma 2.4 gives a link to the general setting; note that the condition (2.3) is satisfied. Fromthis point onward, for G1 and G2 the respective extended marked point process (Ti, Si, ζi)i∈N isalways taken to be that from the proof of Lemma 2.4, see also Example 2.5 below.

Example 2.5. Let

T1 = η, S1 = ∞, ζ1 = (T1, ZT1),

T2 = δ, S2 = ∞, ζ2 = (T2, ZT2),

S2+i = T1 ∧ T2, ζ2+i = (T2+i, ZT2+i), i ∈ N,

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and let T2+i, i ∈ N, be the jump times of the process counting the number of jumps of Z exceptretirement and death. Then according to the proof of Lemma 2.4, cf. Appendix A,

G1t = σ(Ti ≤ t < Si ∩ ζi ∈ A : i ∈ N, A ∈ E) ∨ N ,

G1t− = σ(Ti < t ≤ Si ∩ ζi ∈ A : i ∈ N, A ∈ E) ∨ N ,

for t ≥ 0. The jump times have been permuted so that retirement and death have indices one andtwo, respectively. Consequently, the index of the jump time corresponding to retirement does notcarry information concerning the total number of previous jumps.

In the following, we develop a mathematically sound concept of state-wise prospective reservesin the case of non-monotone information, and we derive so-called stochastic Thiele equationsdescribing the dynamics of state-wise prospective reserves in the presence of non-monotone in-formation. The results are exemplified with non-monotone information given by G1 and G2,respectively, allowing us to discuss current actuarial practice regarding valuation of insuranceliabilities in the presence of (possibly stochastic) retirement.

3. Prospective reserves in the presence of non-monotone information

In the case of monotone information, prospective reserves are so-called optional projections ofaccumulated future payments, suitably discounted. To our knowledge, there appears to be nounifying definition of general state-wise prospective reserves in the actuarial literature; in Nor-berg (1992), state-wise prospective reserves are given implicitly as prospective reserves evaluatedon the relevant event, while Norberg (1996) in principle casts them based on somewhat arbitraryfunctional representations of prospective reserves. The properties of the state-wise prospective re-serves as stochastic processes, including the existence and uniqueness of suitably regular versions,are not investigated. Furthermore, it is unclear from these proposals how to define state-wiseprospective reserves in the presence of non-monotone information.

In this section, we present a sound and fruitful definition of state-wise prospective reserves inthe presence of monotone as well as non-monotone information. In the presence of non-monotoneinformation, the main idea is to take as underlying state process not Z giving the state of theinsured but rather Z giving the state of information. The section is structured as follows. InSubsection 3.1, we introduce so-called state-wise counterparts and reveal the non-triviality ofdeveloping the concept of state-wise prospective reserves. In Subsection 3.2, we follow Christiansen(2020) on optional projections in the presence of non-monotone information, which turns out tobe a fruitful Ansatz for a mathematically sound definition of state-wise quantities. Definitions ofstate-wise prospective reserves are introduced and discussed in Subsection 3.3.

3.1. State-wise counterparts

Suppose that C = (Ct)t≥0 is a sequence of σ-algebras such that

σ(Zt) ∨ N ⊆ Ct ⊆ Ft, t ≥ 0.

Examples include C = G. We define sequences of families of sets Cj = (Ct,j)t≥0, j ∈ S, by

Ct,j = A ∈ Ft− : A ∩ Zt = j ∈ Ct.

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Lemma 3.1. For each (t, j) ∈ [0,∞)×Z the family of sets Ct,j is a σ-algebra. Moreover,

Ct = σ(A ∩ Zt = j : A ∈ Ct,j, j ∈ S)

for any t ≥ 0.

Proof. Follows by standard set-theoretic calculations.

Example 3.2. Consider monotone information F . Then Ft,j = Ft− since Ft− ∨ σ(Zt) ⊆ Ft.

Example 3.3. Consider the setting of Subsection 2.3. By defining

ψ1j (s) := Zs11,...,J(j) + 1η≤s1J+1(j) + (1η≤s,1δ≤s)1J+2(j),

ψ2j (s) := Zs11,...,J(j) + 1δ≤s1J+2(j)

for s ≥ 0 and j ∈ S we find that

G1t,j = σ(ψ1

j (s) : s < t) ∨ N ,

G2t,j = σ(ψ2

j (s) : s < t) ∨ N .

Let Y = (Y (t))t≥0 be a real-valued stochastic process, and suppose that Y (t) is Ct-measurablefor each t ≥ 0. We now define the state-wise counterparts as follows:

Definition 3.4. A family of real-valued stochastic processes (Yj)j∈S = (Yj(t))t≥0,j∈S is said tobe state-wise counterparts to Y if for each (t, j) ∈ [0,∞)× S:

• Yj(t) is Ct,j-measurable,

• 1Zt=jYj(t)a.s.= 1Zt=jY (t).

In general, we suppress the dependency of state-wise counterparts on the specific sequence ofσ-algebras C.

Suppose for the moment that Y G = (Y G(t))t≥0 is the prospective reserve under information G(to be formally defined later on). Then it is intuitively appealing to base the definition of thestate-wise prospective reserves on the state-wise counterparts (Y G

j )j∈S to Y G: they satisfy the

key identity 1Zt=jYGj (t)

a.s.= 1Zt=jY

G(t) and only rely on the information Gt,j , which is theinformation available at time t− that remains available at time t if Zt = j.

For each t ≥ 0 let mt be the sub-probability measure that is uniquely defined on σ(A × j :A ∈ Ct,j , j ∈ S) by

mt(A× j) = mt,j(A) := P (A ∩ Zt = j), A ∈ Ct,j, j ∈ S.

Proposition 3.5. Let Y = (Y (t))t≥0 be a real-valued stochastic process such that Y (t) is integrableand Ct-measurable for each t ≥ 0. Then the state-wise counterparts (Yj)j∈S to Y exist and for

each t ≥ 0 the mapping Ω× S ∋ (ω, j) 7→ Yj(t)(ω) is mt-almost everywhere unique.

Proof. See Appendix A.

The uniqueness of the state-wise counterparts does not extend beyond mt-almost everywherefor fixed t ≥ 0. In other words, viewed as processes the state-wise counterparts are not almostsurely unique and thus not well-defined. Consequently, the definition of state-wise counterparts is

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mathematically flawed and it might therefore be unfortunate to base the definition of state-wiseprospective reserves thereon.

Before we turn the attention to an alternative foundation based on an explicit representation ofoptional projections, we first present some results for the state-wise counterparts that are usefullater.

Define a class of functionals L1(Ω,A, P ) ∋ X 7→ Et,j[X | Ct,j ] by

Et,j[X | Ct,j ] :=E[

X1Zt=j | Ct,j]

E[

1Zt=j | Ct,j] ,

where we impose the convention 0/0 := 0. If P (Zt = j) > 0, it holds that Et,j [X | Ct,j] are versionsof the conditional expectations of Y (t) given Ct,j w.r.t. the probability measure Pt,j given by

Pt,j(A) =P (A ∩ Zt = j)

P (Zt = j), A ∈ A,

cf. Exercise 34.4(a) of Billingsley (1994).Based on similar techniques as in the proof of Proposition 3.5, one can then show that

Yj(t)a.s.= Et,j[Y (t) | Ct,j ] . (3.1)

This provides an explicit representation of the state-wise counterparts.We are now ready to derive the following law of iterated expectations:

Lemma 3.6. Let X ∈ L1(Ω,A, P ). Then for each (t, j) ∈ [0,∞) × S:

Et,j [E[X | Ct] | Ct,j ]a.s.

= Et,j[X | Ct,j ] .

Proof. See Appendix A.

When Ct,j is generated by Ft,j = ft,j((Zs)0≤s<t) added null sets N with ft,j some measurablefunction, it can be shown that

Et,j [Y (t) | Ct,j ]a.s.= E[Y (t) |Ft,j , Zt = j] , (3.2)

where the latter refers to path-wise integration w.r.t. the conditional distribution of Y (t) given(Ft,j , Zt) and, further, evaluated in Ft,j(ω), j. This provides an alternative explicit representa-tion of the state-wise counterparts. Rewrites in the spirit of (3.2) are typical and occur frequentlyand opaquely in the remainder of the paper.

3.2. Optional projections and state-wise quantities

Let Y = (Y (t))t≥0 be a real-valued stochastic process such that Y (t) is integrable for each t ≥ 0.If there exists an almost surely unique process X = (X(t))t≥0 such that for each t ≥ 0,

X(t) = E [Y (t) | Gt]

almost surely, then we denote Y G := X as the optional projection of Y with respect to G.In the following we calculate conditional expectations given (ζx, Txy, ζxy), x, y ∈ S, x 6= y. We

throughout assume that they are defined as path-wise integrals with respect to arbitrary but fixed

11

regular conditional distributions P (· | ζx, Txy, ζxy). For a cadlag or caglad process Y = (Y (t))≥0

with finite expected variation on compacts, let

YGx (t) :=

E[Ix(t)Y (t) | ζx]

E[Ix(t) | ζx], t ≥ 0,

YG−x (t) :=

E[Ix(t−)Y (t) | ζx]

E[Ix(t−) | ζx], t > 0,

YGxx(t) :=

E[Ix(t−)Ix(t)Y (t) | ζx]

E[Ix(t−)Ix(t) | ζx], t > 0,

YG−xy (t, e) :=

E[Ix(t−)Y (t) | ζx, Txy = t, ζxy = e]

E[Ix(t−) | ζx, Txy = t, ζxy = e], x 6= y, e ∈ Exy, t > 0,

YGxy(t, e) :=

E[Iy(t)Y (t) | ζy, Txy = t, ζxy = e]

E[Iy(t) | ζy, Txy = t, ζxy = e], x 6= y, e ∈ Exy, t > 0,

(3.3)

which are almost surely unique processes, cf. the discussion between Theorem 4.2 and Proposi-tion 4.3 of Christiansen (2020). The above state-wise quantities refer to the state of informationand changes in the state of information, rather than the state of the insured. In Subsection 3.3we interpret these state-wise quantities when Y describes the accumulated future payments. Thefollowing proposition helps us in this regard.

Proposition 3.7. Let Y be a cadlag or caglad process with finite expected variation on compacts.

For each t > 0 we almost surely have

Ix(t)YGx (t) = Ix(t)E[Y (t) | Gt],

Ix(t−)YG−x (t) = Ix(t−)E[Y (t) | Gt−],

Ix(t−)YGxx(t) = Ix(t−)E[Y (t) | Gt−,Zt = x],

Ix(t)YGxx(t) = Ix(t)E[Y (t) | Gt,Zt− = x],

Ix(t−)YG−xy (t, e) = Ix(t−) E[Y (t) | Gt−, Txy = t, ζxy = e],

Iy(t)YGxy(t, e) = Iy(t) E[Y (t) | Gt, Txy = t, ζxy = e].

Proof. See Proposition 4.3 and the proof of Theorem 4.2 in Christiansen (2020).

Note that for all t > 0 and x ∈ S, by Proposition 3.7 and (2.5), the latter which essentiallyallows us to replace Gt− by ζx on Zt− = x,

Ix(t−)YGxx(t) = Ix(t−)E[Y (t) | Gt−,Zt = x] = Ix(t−)

E[Ix(t)Y (t) | ζx]

E[Ix(t) | ζx]= Ix(t−)YG

x (t) (3.4)

almost surely.The state-wise quantities YG

x allow for a rather explicit characterization of the optional projec-tion Y G :

Proposition 3.8. Let Y = (Y (t))t≥0 be a cadlag process with with finite expected variation

on compacts. Then the optional projection Y G of Y exists and has the almost surely unique

representation

Y G(t) =∑

x∈S

Ix(t)YGx (t), t ≥ 0.

For each x ∈ S the process t 7→ Ix(t)YGx (t) has a cadlag modification with paths of finite variation

on compacts.

12

Proof. See Theorem 4.2 in Christiansen (2020).

The following corollary shows that the path properties of the optional projection Y G are passedon to the state-wise quantities of interest. It ensures all later applications of e.g. integration byparts to be feasible.

Corollary 3.9. Let Y = (Y (t))t≥0 be a cadlag process with finite expected variation on compacts.

Then the processes

[0,∞) ∋ t 7→ 1Zt=jYGj (t), (0,∞) ∋ t 7→ 1Zt−=jY

Gj (t),

and the process

(0,∞) ∋ t 7→ Ix(t−)YGx (t)

almost surely have cadlag paths of finite variation on compacts.

Proof. See Appendix A.

In the special case of monotone information, we now establish a more direct relation betweenthe different concepts of state-wise quantities. Setting (Ti, Si, ζi)i∈N := (τi,∞, (Zτi , τi))i∈N werecover the filtration F = G. In this case, let

YF−

jk (t) := 1Zt−=j

∑

x,y∈Sx 6=y

Ix(t−)YF−xy (t, (k, t)),

YF−

jj (t) := 1Zt−=j

∑

x∈S

Ix(t−)YFxx(t)

(3.5)

for j, k ∈ S, j 6= k, and t > 0.

Remark 3.10. In case of (Ti, Si, ζi)i∈N := (τi,∞, (Zτi , τi))i∈N, only those indicator processes Ixare different from constantly zero that have an x of the form x = 1, . . . , n ∈ S for some n ∈ N0;here we define 1, . . . , n as the empty set in case of n = 0. In particular, we have

Ix(t−) = 1τn<t≤τn+1 if x = 1, . . . , n

for t > 0 and with τ0 := 0. Moreover, the stopping times Txy are only then different fromconstantly infinity if x and y are of the form x = 1, . . . , n, y = 1, . . . , n + 1, n ∈ N0. Inparticular, for each t > 0 we almost surely have

YF−

jk (t) = 1Zt−=j

∞∑

n=0

1τn<t≤τn+1E[Y (t) | (Zτ1 , τ1), . . . , (Zτn , τn), (Zτn+1, τn+1) = (k, t)],

YF−

jj (t) = 1Zt−=j

∞∑

n=0

1τn<t≤τn+1

E[1Zt=jY (t) | (Zτ1 , τ1), . . . , (Zτn , τn)]

E[1Zt=j | (Zτ1 , τ1), . . . , (Zτn , τn)]

(3.6)

for j, k ∈ S, j 6= k.

In the presence of monotone information, the following result relates the state-wise counterpartsto the state-wise quantities introduced by (3.3).

13

Proposition 3.11. Let Y = (Y (t))t≥0 be a cadlag process with finite expected variation on

compacts. Denote by Y F the corresponding optional projection and by (Y Fj )j∈S the state-wise

counterparts to Y F . At each time t > 0 it almost surely holds that

Y Fj (t) = Y

F−

jj (t) +∑

k∈Sj 6=k

YF−

kj (t)

for j ∈ S, where YF−

jj and YF−

kj , k 6= j are almost surely unique predictable processes defined by

(3.5).

Proof. See Appendix A.

In the following, the notation Y Fj always refers to the modification given by Proposition 3.11.

Insisting on this essentially solves the issue of well-definedness of the state-wise counterparts inthe presence of monotone information. In the general case, where we allow for non-monotoneinformation, the issue persists.

Example 3.12. Consider the accumulated payments B, which is an F-adapted cadlag processwith finite expected variation on compacts; in particular, BF = B. Proposition 3.11 yields

B(t) =∑

j∈S

1Zt=jBFj (t)

=∑

j∈S

1Zt=jBF−

jj (t) +∑

j,k∈Sj 6=k

1Zt=jBF−

kj (t)

almost surely for all t > 0. Recall that BF−

jj (t) = 1Zt−=jBF−

jj (t) and BF−

jk (t) = 1Zt−=jBF−

jk (t)for all j, k ∈ S, j 6= k. By applying integration by parts and rearranging the terms, one then finds

B(dt) =∑

j∈S

1Zt−=jBF−

jj (dt) +∑

j,k∈Sj 6=k

(BF−

jk (t)−BF−

jj (t))Njk(dt) (3.7)

almost surely. This recovers the classic decomposition into sojourn payments and transitionpayments in the following sense. Suppose the accumulated payments B are defined as

B(dt) =∑

j∈S

1Zt−=jBj(dt) +∑

j,k∈Sj 6=k

bjk(t)Njk(dt),

where the cumulative sojourn payments Bj shall be F-predictable cadlag processes with finite ex-pected variation on compacts and the transition payments bjk shall be bounded and F-predictable

processes. By calculating BF−

jj and BF−

jk in (3.7) explicitly and comparing the results with thedefinition of B, one can show that

1Zt−=jBj(dt) = 1Zt−=jBF−

jj (dt)

almost surely for j ∈ S and for each t > 0,

1Zt−=jbjk(t) = 1Zt−=j

(

BF−

jk (t)−BF−

jj (t))

14

almost surely for j, k ∈ S, j 6= k. By defining the process β = (β(t))t≥0 via

β(t) =∑

j,k∈Sj 6=k

bjk(t)∆Njk(t), t ≥ 0,

which equals the difference of a cadlag and a caglad process, we can alternatively recover thetransition payments via the representation

1Zt−=jbjk(t) = 1Zt−=jβF−

jk (t),

which holds almost surely for all t > 0 and j, k ∈ S, j 6= k.

3.3. State-wise prospective reserves

In the previous two subsections, we have introduced a range of state-wise concepts and quantities,including the state-wise counterparts, and we have studied their interrelation – in particular in thepresence of monotone information. Building on this, we now turn our attention to mathematicalsound definitions of state-wise prospective reserves. In the presence of monotone information, thedefinition bases on the concept of state-wise counterparts and refers to the state of the insured,while in the presence of non-monotone information, we rely on the state-wise quantities appear-ing in the explicit characterization of optional projections; these quantities refer to the state ofinformation rather than the state of the insured.

Consider a deterministic bank account κ : [0,∞) 7→ (0,∞) assumed measurable, cadlag, andof finite variation on compacts, with initial value κ(0) = 1. Denote with v the correspondingdiscount function given by

[0,∞) ∋ t 7→ v(t) =1

κ(t).

Denote from this point on by Y = (Y (t))t≥0 the accumulated future payments, suitably dis-counted, given by

Y (t) =

∫

(t,∞)

κ(t)

κ(s)B(ds).

Note that Y has cadlag sample paths of finite variation on compacts. We further suppose thatY (t) has finite expected variation on compacts. This is for example the case if κ is bounded awayfrom zero.

The prospective reserve under possibly non-monotone information is the almost surely uniqueoptional projection Y G = (Y G(t))t≥0 of Y w.r.t. G satisfying for each t ≥ 0

Y G(t) = E [Y (t) | Gt] = E

[

∫

(t,∞)

κ(t)

κ(s)B(ds)

∣

∣

∣

∣

∣

Gt

]

(3.8)

almost surely. This definition is consistent with the one proposed in Norberg (1991). State-wiseprospective reserves are now defined as follows:

Definition 3.13. For j ∈ S the classic state-wise prospective reserve in insured state j is the notnecessarily unique process Y G

j = (Y Gj (t))t≥0, where (Y

Gj )j∈S are the state-wise counterparts to the

15

prospective reserve Y G . For x ∈ S, the non-classic state-wise prospective reserve in information

state x is the almost surely unique process YGx = (YG

x (t))t≥0 given by

YGx (t) =

E[Ix(t)Y (t) | ζx]

E[Ix(t) | ζx]

for t ≥ 0.

In the following we shall follow the conventions of the literature and write (Vj)j∈S for theclassic state-wise prospective reserves in the presence of monotone information G = F . Similarly,we write V for the prospective reserve in the presence of monotone information.

Note that for each t ≥ 0, j ∈ S, and x ∈ S, it almost surely holds that

1Zt=jYGj (t) = 1Zt=jY

G(t),

Ix(t)YGx (t) = Ix(t)Y

G(t),

cf. Definition 3.4 and Proposition 3.7. The proposed explicit definitions are therefore consistentwith the implicit definition in the presence of monotone information put forward by Norberg(1992).

By an application of the law of iterated expectations, cf. Remark 3.6, and the identity (3.1),we can for each t ≥ 0 cast the classic state-wise prospective reserves as

Y Gj (t)

a.s.= E

[

∫

(t,∞)

κ(t)

κ(s)B(ds)

∣

∣

∣

∣

∣

Gt,j , Zt = j

]

, j ∈ S. (3.9)

Example 3.14. Consider the case of monotone information F , when by Example 3.2 we haveFt,j = Ft−. It follows that for each t ≥ 0 and j ∈ S,

Vj(t)a.s.= E

[

∫

(t,∞)

κ(t)

κ(s)B(ds)

∣

∣

∣

∣

∣

(Zs)0≤s<t, Zt = j

]

,

cf. (3.9) and (3.2).

Example 3.15. Consider the framework of Subsection 2.3 with non-monotone information Gi,i ∈ 1, 2, when by Example 3.3 we have Gi

t,j = σ(ψij(t)) ∨ N . In the presence of non-monotone

information Gi, i ∈ 1, 2, we then for each t ≥ 0 and j ∈ S have

Y Gi

j (t)a.s.= E

[

∫

(t,∞)

κ(t)

κ(s)B(ds)

∣

∣

∣

∣

∣

(ψij(s))0≤s<t, Zt = j

]

,

cf. (3.9). For example,

Y G1

J+1(t)a.s.= E

[

∫

(t,∞)

κ(t)

κ(s)B(ds)

∣

∣

∣

∣

∣

Ut, Zt = J + 1

]

,

where U = (Ut)t≥0 is the duration process associated with Z.Note that for each t ≥ 0,

Y Gi

j (t)a.s.= Vj(t)

16

for j ∈ 1, . . . , J, while applying (3.2), (3.3), and the constructions of G1 and G2 according tothe proof of Lemma 2.4, yields

Y Gi

J+1(t)a.s.= YGi

1(t),

Y Gi

J+2(t)a.s.= 1η≤tY

Gi

1,2(t) + 1η>tYGi

2(t).

In the following, (Y Gi

j )j∈S always refers to the modifications given by the above identities. Insistingon this ensures the classic state-wise prospective reserves to be well-defined in the presence of non-monotone information Gi.

As already discussed in Subsections 3.1-3.2, the state-wise counterparts are as a rule not well-defined as stochastic processes, since they are defined up to null-sets for an uncountable number oftime points. In the presence of monotone information, G = F , we insist on taking the modificationgiven by Proposition 3.11, which solves the problem of well-definedness, and in the followingsection we show how the concept of classic state-wise prospective reserves is sufficient to studydynamics of state-wise prospective reserves under monotone information. In the presence of non-monotone information, the classic state-wise prospective reserves are not well-defined as stochasticprocesses. Furthermore, as we show in the following section, the concept of non-classic state-wiseprospective reserves, as well as the additional state-wise quantities given by (3.3), is necessaryto study the dynamics of state-wise prospective reserves under non-monotone information. Todevelop the general theory of stochastic Thiele equations, we thus focus on the non-classic state-wise prospective reserves, which refer to the state of information. Still, when meaningful andrelevant for specific instances of information, cf. Example 3.15, we cast the results in terms of themore intuitively appealing classic state-wise prospective reserves, which refer to the state of theinsured.

In addition to the classic and non-classic state-wise prospective reserves, the additional state-wise quantities given by (3.3) prove useful. Based on Proposition 3.8 and Proposition 3.7, for

each x, y ∈ S, x 6= y, we interpret the state-wise quantities YGxx, Y

Gxy, and Y

G−xy as follows:

• YGxx(t) is the prospective reserve for staying in information state x at time t: if in information

state x at time t− or time t, what one would set aside in case no change in informationstate occurs at time t,

• YGxy(t, e) is the backward prospective reserve at transition from information state x to infor-

mation state y with information change e: if in information state y at time t, what one wouldset aside in case a change from information state x occurred with change in information eat exactly time t,

• YG−xy (t, e) is the forward prospective reserve at transition from information state x to y with

information change e: if in information state x at time t−, what one would set aside in casea change to information state y occurs with change in information e at exactly time t.

4. Dynamics of state-wise prospective reserves

In this section, we present the main results of the paper by deriving so-called stochastic Thieleequations describing the dynamics of state-wise prospective reserves in the presence of non-monotone information. In principle, our method is based on the infinitesimal approach introducedand developed by Christiansen (2020) and relies on the explicit infinitesimal martingale repre-sentation theorem (see Theorem 6.1 and Theorem 7.1 in Christiansen, 2020). In comparison,

17

stochastic Thiele equations in the presence of monotone information are closely related to theclassic martingale representation theorem, see e.g. Norberg (1992) and Christiansen and Djehiche(2019).

In Subsection 4.1, we present and derive so-called infinitesimal forward/backward compensatorsdescribing the systematic part of the development of the state of information and the payments.Generalized stochastic Thiele equations are derived and interpreted in Subsection 4.2. Finally, inSubsection 4.3 we impose the specific framework of Subsection 2.3 with non-monotone informationrelated to information discarding upon and after stochastic retirement and derive stochastic Thieleequations and – in the presence of certain intertemporal dependency structures – Feynman-Kacformulas exemplifying our results.

In the remainder of the paper, we generally suppose that

B(dt) =∑

j∈S

1Zt−=jbj(t)µ(dt) +∑

j,k∈Sj 6=k

bjk(t)Njk(dt),

where bj and bjk are F-predictable bounded processes and the measure µ is a sum of the Lebesgue-measure m and a countable number of Dirac-measures (ǫtn)n∈N:

µ(A) = m(A) +∞∑

n=1

ǫtn(A), A ∈ B([0,∞)),

for deterministic time points 0 ≤ t1 < t2 < ... that are increasing to infinity (i.e. there are at mosta finite number of such time points on each compact interval).

4.1. Infinitesimal compensators

The so-called compensator λxy of the random counting measure νxy is the unique F-predictablerandom measure such that the difference [0,∞) ∋ t 7→ νxy([0, t] × A) − λxy([0, t] × A) is anF-martingale for each A ∈ B(Exy). In particular, we have

λxy((0, t] ×A) = limn→∞

∑

Tn

E[νxy((tk, tk+1]×A) | Ftk ] (4.1)

almost surely for each t > 0, where (Tn)n∈N is any increasing sequence (i.e. Tn ⊂ Tn+1 for alln) of partitions 0 = t0 < · · · < tn = t of the interval [0, t] such that |Tn| := maxtk − tk−1 :k = 1, . . . , n → 0 for n → ∞. Christiansen (2020) expands this property to the non-motonone

information G and denotes the random measures γG−xy and γGxy defined by

γG−xy ((0, t] ×A) = lim

n→∞

∑

Tn

E[νxy((tk, tk+1]×A) | Gtk ], t > 0, A ∈ B(Exy),

γGxy((0, t] ×A) = limn→∞

∑

Tn

E[νxy((tk, tk+1]×A) | Gtk+1], t > 0, A ∈ B(Exy),

as infinitesimal forward compensator (IF-compensator) and infinitesimal backward compensator(IB-compensator) of νxy with respect to G, given that the limits exist for all t > 0 almost surely.

In the special case of monotone information G = F the IF-compensator equals the classiccompensator and the IB-compensator equals the counting measure itself, i.e. γ

F−xy = λxy and

γFxy = νxy almost surely.

18

Proposition 4.1. For each x, y ∈ S, x 6= y, the IF-compensator γG−xy and the IB-compensator

γGxy of νxy exist and satisfy for each A ∈ B(Exy) and all t > 0

γG−xy ((0, t] ×A) =

∫

(0,t]×A

Ix(s−)

E[Ix(s−) | ζx]P ((Txy, ζxy) ∈ ds× de | ζx),

γGxy((0, t]×A) =

∫

(0,t]×A

Iy(s)

E[Iy(s) | ζy]P ((Txy, ζxy) ∈ ds× de | ζy)

almost surely.

Proof. See Proposition 5.1 and Theorem 5.2 in Christiansen (2020).

Denote by b the sojourn payment rate given by

b(t) =∑

j∈S

1Zt−=jbj(t), t > 0,

and denote by β the transition payments given by

β(t) =∑

j,k∈Sj 6=k

bjk(t)∆Njk(t), t > 0.

Proposition 4.2. The payment process B has an IF-compensator CG−

B with respect to G of the

form

CG−

B (dt)a.s.

=∑

x∈S

Ix(t−)bG−x (t)µ(dt) +

∑

x,y∈Sx 6=y

∫

Exy

βG−xy (t, e) γ

G−xy (dt× de),

where bG−x and β

G−xy are the processes defined from b and β by the second and fourth line in (3.3),

respectively.

Proof. See Theorem 5.2 and Example 7.2 in Christiansen (2020). Note that (2.4) holds and thatβ can be decomposed into a sum of a cadlag and a caglad process both with finite expectedvariation on compacts.

Applying similar techniques as in the proof of Proposition 4.1 and the proof of Proposition 4.2,one can show that if for all t > 0 each bj(t) and bjk(t) is Gt−-measurable, then

CG−

B (dt)a.s.=∑

j∈S

1Zt−=jbj(t)µ(dt) +∑

j,k∈Sj 6=k

bjk(t) ΓG−

jk (dt),

where ΓG− are the IF-compensators of the multivariate counting process N (associated with Z)w.r.t. G.

In general, we thus interpret bG−x as the (G-averaged) sojourn payments in information state

x ∈ S and βG−xy (·, e) as the (G-averaged) transition payment for a change in information e from

information state x to information state y.

19

4.2. Stochastic Thiele equations

We are now ready to present stochastic differential equations describing the dynamics of thenon-classic state-wise prospective reserves (YG

x )x∈S in the presence of general non-monotone in-formation G:

Theorem 4.3 (Generalized stochastic Thiele equation). The non-classic state-wise prospective

reserves (YGx )x∈S almost surely satisfy the stochastic differential equation

0 =∑

x∈S

Ix(t−)

(

YGx (dt)− YG

x (t−)κ(dt)

κ(t−)+ bG−

x (t)µ(dt) +∑

y:y 6=x

∫

Exy

RG−(t, x, y, e) γG−xy (dt× de)

−∑

y:y 6=x

∫

Eyx

RG(t, y, x, e) γGyx(dt× de)

)

,

(4.2)

where for x, y ∈ S, x 6= y,

RG−(t, x, y, e) := βG−xy (t, e) + YG−

xy (t, e) −YGxx(t),

RG(t, y, x, e) := YGyx(t, e)− YG

xx(t).

Remark 4.4. Due to (3.4), we could equally likely cast RG− and RG via

RG−(t, x, y, e) := βG−xy (t, e) + YG−

xy (t, e) − YGx (t),

RG(t, y, x, e) := YGyx(t, e)− YG

x (t).

In the following, we prefer this representation.

In the presence of monotone information G = F , starting from Theorem 4.3 one can derivethe following stochastic differential equations describing the dynamics of the classic state-wiseprospective reserves (Vj)j∈S:

Corollary 4.5 (Classic stochastic Thiele equation). The classic state-wise prospective reserves

(Vj)j∈S almost surely satisfy the stochastic differential equation

0 =∑

j∈S

1Zt−=j

(

Vj(dt)− Vj(t−)κ(dt)

κ(t−)+ bj(t)µ(dt) +

∑

k:k 6=j

Rjk(t)Λjk(dt)

)

, (4.3)

where Rjk(t) := bjk(t) + Vk(t)− Vj(t) are the classic sum at risks and where Λjk := ΓF−

jk are the

classic F-compensators of the multivariate counting process N .

Before we present the proofs of Theorem 4.3 and Corollary 4.5, we first provide an interpretationof the results. In the presence of monotone information, Corollary 4.5 yields stochastic differentialequations that are directly comparable to the stochastic Thiele equations of Norberg (1992, 1996).In Norberg (1992, 1996), the F-compensators Λ of N are assumed to admit densities w.r.t. theLebesgue-measure, and the result is derived by suitably applying the martingale representationtheorem and identifying the integrands. The method of the present paper, while extended toalso cover the non-monotone case, is based on a suitable application of the explicit infinitesimalmartingale representation theorem. In particular, Corollary 4.5 can also be derived directly from

20

the classic martingale representation theorem following Christiansen and Djehiche (2019); in thiscase, the restriction to slightly less general payments, cf. beginning of Section 4, is not necessary.

The stochastic differential equation of Theorem 4.3 is in a twofold manner fundamentally dif-ferent from the stochastic Thiele equation in the presence of monotone information. Firstly, thesum at risks appearing in the term involving the IF-compensators, which correspond to ordinarycompensators in the presence of monotone information, take a different form. Rather than be-ing the difference of two state-wise prospective reserves added the relevant transition payment,it involves the difference of the forward state-wise prospective reserve and the current ordinarystate-wise prospective reserve added relevant transition payment. In the presence of monotoneinformation, we can show that the forward state-wise prospective reserve can be replaced by a rel-evant ordinary state-wise prospective reserve, but this is not necessarily the case in the presenceof non-monotone information. Here the possibility of information discarding entails a possibleimprovement in the accuracy of the reserving by utilizing the information available at time t−and time t, rather than utilizing only the information available at time t.

Secondly, the stochastic differential equation of Theorem 4.3 contains an additional term in-volving the IB-compensators. In the presence of monotone information, we can show that thisterm is zero. It is the backward looking equivalent of the term involving the IF-compensators.Based on the information currently available, the term adjusts the dynamics to take into accountthe possibility that information discarding has just occurred.

In Subsection 4.3, we derive and interpret stochastic Thiele equations in the presence of spe-cific examples of non-monotone information related to stochastic retirement. We refer to thissubsection for further interpretation and discussion of the general results.

Proof of Theorem 4.3. Analogously to Proposition 4.2, one can show that the discounted paymentprocess B given by B(0) = B(0) and B(dt) := v(t)B(dt) admits the IF-compensator

CG−

B(dt) =

∑

x∈S

Ix(t−)v(t) bG−x (t)µ(dt) +

∑

x,y∈Sx 6=y

∫

Exy

v(t)βG−xy (t, e) γ

G−xy (dt× de).

According to Theorem 7.1 in Christiansen (2020), the process [0,∞) ∋ t 7→ Y (t) = v(t)Y (t)almost surely satisfies the equation

Y G(dt) = −CG−

B(dt) +

∑

x,y∈Sx 6=y

∫

Exy

v(t)(

YG−xy (t, e) − YG

xx(t))

(νxy − γG−xy )(dt× de)

−∑

x,y∈Sx 6=y

∫

Exy

v(t)(

YGxy(t, e)− YG

xx(t))

(νxy − γGxy)(dt× de).

On the other hand, by applying integration by parts on Y (t)a.s.=∑

x∈S Ix(t)YGx (t) and using

Y G a.s.= v(t)Y G, we can show that

Y G(dt)a.s.=∑

x∈S

Ix(t−)YGx (dt) +

∑

x,y∈Sx 6=y

v(t)(

YGy (t)−YG

x (t))

νxy(dt× Exy).

Thus, by equating the latter two equations and rearranging the terms, while using (3.4), the fact

that γG−xy (dt× de) = Ix(t−)γ

G−xy (dt× de) and γGyx(dt× de) = Ix(t)γ

Gyx(dt× de) almost surely, and

21

the equation Ix(t) = Ix(t−)Ix(t) + 1Zt− 6=xIx(t), we obtain

0a.s.=∑

x∈S

Ix(t−)

(

YGx (dt) + v(t)bG−

x (t)µ(dt) +∑

y:y 6=x

∫

Exy

v(t)RG−(t, x, y, e) γG−xy (dt× de)

−∑

y:y 6=x

∫

Eyx

v(t)RG(t, y, x, e) γGyx(dt× de)

)

,

−∑

x,y∈Sx 6=y

v(t)(

YG−

xy (t, e) −YGxy(t, e)

)

νxy(dt× Exy)

+∑

x,y∈Sx 6=y

v(t)(

YGxx(t)− YG

yy(t))

νxy(dt× Exy)

−∑

x,y∈Sy 6=x

∫

Eyx

v(t)(

YGyx(t, e)− YG

xx(t))

1Zt− 6=xIx(t) γGyx(dt× de)

+∑

x,y∈Sx 6=y

v(t)(

YGy (t)− YG

x (t))

νxy(dt× Exy).

The third line equals zero because of (6.7) in Christiansen (2020). The forth, fifth and sixth linetogether equal

∑

x,y∈Sx 6=y

v(t)(YGy (t)− YG

yy(t)) νxy(dt× Exy)

−∑

x,y∈Sy 6=x

(∫

Eyx

v(t)(

YGyx(t, e) − YG

xx(t))

(

∑

z:z 6=x

νzx(t × Ezx)

)

γGyx(dt× de)

)

+∑

x,y∈Sx 6=y

v(t)(YGxx(t)− YG

x (t)) νxy(dt× Exy)

=∑

x,y∈Sx 6=y

v(t)(YGx (t)− YG

xx(t)) νyx(dt×Eyx)

−∑

x,y∈Sy 6=x

(

∑

z:z 6=x

∫

Ezx

v(t)(

YGzx(t, e)− YG

xx(t))

γGzx(t × de)

)

νyx(dt× Eyx)

+∑

x,y∈Sx 6=y

v(t)(YGxx(t)− YG

x (t)) νxy(dt× Exy)

almost surely, because∑

z:z 6=x νzx(t×Ezx) is almost surely non-zero only at finitely many time

points. The last line equals zero since Ix(t−)(YGxx(t)−YG

x (t)) = 0 according to (3.4). The remain-

22

ing two lines also add up to zero since, by applying Proposition 3.7, (2.5), and Proposition 4.1,

Ix(t)YGx (t) = Ix(t)E[Y (t) | ζx,Zt = x]

= Ix(t)E[Y (t)Ix(t−) | ζx,Zt = x] + Ix(t)E

[

Y (t)∑

z:z 6=x

Iz(t−)

∣

∣

∣

∣

ζx,Zt = x

]

= Ix(t)E[Y (t) | ζx,Zt = x,Zt− = x] E[Ix(t−) | ζx,Zt = x]

+∑

z:z 6=x

∫

Ezx

E[Y (t) | ζx,Zt = x, Tzx = t, ζzx = e] γGzx(t × de)

= Ix(t)YGxx(t)

(

1−∑

z:z 6=x

γGzx(t × Ezx)

)

+∑

z:z 6=x

∫

Ezx

YGzx(t, e) γ

Gzx(t × de)

almost surely. All in all, we have

0 =∑

x∈S

Ix(t−)

(

YGx (dt) + v(t)bG−

x (t)µ(dt) +∑

y:y 6=x

∫

Exy

v(t)RG−(t, x, y, e) γG−xy (dt× de)

−∑

y:y 6=x

∫

Exy

v(t)RG(t, x, y, e) γGyx(dt× de)

)

.

Now apply integration by parts on YGx (t) = v(t)YG

x (t) and rearrange the terms in order to end upwith the statement of the theorem.

Proof of Corollary 4.5. By setting (Ti, Si, ζi)i∈N = (τi,∞, (τi, Zτi))i∈N we obtain G = F such that(YF

x )x∈S satisfy (4.2) almost surely. Recall that γFyx = νyx, when

Ix(t−)∑

y:y 6=x

∫

Eyx

RF (t, x, y, e) γFyx(dt× de) = 0

almost surely. By Remark 3.10 and starting from (4.2), similar arguments as in the proof ofProposition 3.11 yield the following stochastic differential equations:

∞∑

n=0

1Zτn=j1τn<t≤τn+1YF1,...,n(dt)

a.s= 1Zt−=j

(

Vj(t−)κ(dt)

κ(t−)− bj(t)µ(dt)−

∑

k:k 6=j

(

bjk(t) + Vk(t)− Vj(t))

ΓF−

jk (dt)

)

for j ∈ S. By tedious yet straightforward calculations, it is possible to show that

∞∑

n=0

(

Vj(t)− YF1,...,n

)

d(

1Zτn=j1τn≤t<τn+1

) a.s.= 0, j ∈ S,

which implies

∞∑

n=0

1Zτn=j1τn<t≤τn+1YF1,...,n(dt)

a.s.= 1Zt−=jVj(dt), j ∈ S,

by an application of integration by parts. Collecting results establishes the desired result.

23

In the case where the payments B themselves depend on the prospective reserve V , the (stochas-tic) Thiele equations rather than (3.8) might serve as definition for the prospective reserve V ,see e.g. Djehiche and Lofdahl (2016) and Christiansen and Djehiche (2019). In the presence ofmonotone information, this point of view is encapsulated by the following result.

Proposition 4.6. Let there be a maximal contract time n <∞, i.e. each bj and bjk is constantly

zero on the interval (n,∞). Suppose that Wj , j ∈ S, are F-predictable bounded processes such

that [0,∞) ∋ t 7→ 1Zt=jWj(t) almost surely has cadlag paths for all j ∈ S. If Wj, j ∈ S, satisfythe stochastic differential equations

0 = 1Zt−=j

(

Wj(dt)−Wj(t−)κ(dt)

κ(t−)+ bj(t)µ(dt) +

∑

k:k 6=j

(bjk(t)+Wk(t)−Wj(t))Λjk(dt)

)

(4.4)

with terminal condition Wj(n) = 0, j ∈ S, then WZt(t) = V (t) almost surely for all t ∈ [0, n].

Proof. Since [0,∞) ∋ t 7→ 1Zt=jWj(t) almost surely has cadlag paths for all j ∈ S, the stochasticdifferential equations imply that [0,∞) ∋ t 7→ 1Zt=jWj(t) and (0,∞) ∋ t 7→ 1Zt−=jWj(t)almost surely have cadlag paths of finite variation for all j ∈ S, cf. proof of Corollary 3.9. Byapplying integration by parts and the stochastic differential equations for the processesWj, j ∈ S,we obtain

d(

v(t)WZt(t))

=∑

j∈S

1Zt−=j

(

v(t)Wj(dt)−Wj(t−)v(t)κ(dt)

κ(t−)

)

+∑

j,k∈Sj 6=k

v(t)(Wk(t)−Wj(t))Njk(dt)

= −v(t)B(dt) +∑

j,k∈Sj 6=k

v(t)(bjk(t) +Wk(t)−Wj(t))(Njk − Λjk)(dt)

almost surely. Since each [0,∞) ∋ t 7→ bjk(t) +Wk(t)−Wj(t) is F-predictable and bounded, thelast term is an F-martingale. Thus, we obtain

v(t)WZt(t) = E

[

v(t)∑

j∈S

1Zt=jWj(t)

∣

∣

∣

∣

Ft

]

= E

[

v(t)

∫

(t,n]

κ(t)

κ(s)B(ds)

∣

∣

∣

∣

Ft

]

= v(t)V (t)

almost surely for all t ∈ [0, n]. Noting v > 0 completes the proof.

4.3. Examples

In this subsection, we consider the framework of stochastic retirement from Subsection 2.3 andthe non-monotone information given by G1 and G2. The time of retirement and death are givenby the hitting times η and δ, respectively. Recall that G1 corresponds to the case where uponretirement or death the insurer discards the previous health records of the insured, while G2 evenkeeps no record on the time of retirement.

In Subsection 4.3.1, we present some auxiliary results characterizing the relevant IF- and IB-compensators and state-wise quantities. Stochastic Thiele equations are then derived in Subsec-tion 4.3.2 using the general theory developed in Subsection 4.2. Finally, in Subsection 4.3.3 wespecialize the inter-temporal dependency structure, derive Feynman-Kac formulas, and relate theresults to actuarial practice.

24

4.3.1. Preliminaries

Denote for j, k ∈ S, k 6= j, by ΓGi−

jk and ΓGi

jk the IF- and IB-compensator of Njk, respectively, w.r.t.

Gi, i = 1, 2. Recall that Λ denotes the classic F-compensators of N . The following result givesan explicit characterization of the relevant IF- and IB-compensators of N w.r.t. G1 and G2.

Proposition 4.7. For all t > 0 we almost surely have

ΓG1−jk (t) = Γ

G2−

jk (t) = Λjk(t), j ∈ 1, . . . , J, k ∈ S \ j,

ΓG1

jk (t) = ΓG2

jk (t) = Njk(t), either j ∈ S, k ∈ 1, . . . , J \ j or j = J + 1, k = J + 2,

ΓG1−jk (t) =

∫

(0,t]

1η<s≤δ

P (δ ≥ s | η)P (δ ∈ ds | η), j = J + 1, k = J + 2,

ΓG2−jk (t) =

∫

(0,t]

1η<s≤δ

P (η < s ≤ δ)P (δ ∈ ds, Zδ− = J + 1), j = J + 1, k = J + 2,

ΓG1

jk (t) =

∫

(0,t]P (Zη− = j | η = s)

J∑

ℓ=1

Nℓk(ds), j ∈ 1, . . . , J, k = J + 1,

ΓG2

jk (t) =

∫

(0,t]

1η≤s<δ

P (η ≤ s < δ)P (η ∈ ds, Zη− = j), j ∈ 1, . . . , J, k = J + 1,

ΓG1

jk (t) = ΓG2

jk (t) =

∫

(0,t]

P (Zδ− = j | δ = s)

P (Zδ− 6= J + 1 | δ = s)

J∑

ℓ=1

Nℓk(ds), j ∈ 1, . . . , J, k = J + 2.

All remaining IF- and IB-compensators of N equal zero almost surely .

Sketch of proof. Calculate the IF-compensator γGi−

xy and the IB-compensator γGixy of νxy from

Proposition 4.1 and use the construction of G1 and G2 according to the proof of Lemma 2.4.

In the following, (Y Gi

j )j∈S refers to the modification of the classic state-wise prospective reserves

w.r.t. Gi presented in Example 3.15. The next result provides a characterization of the remainingkey terms appearing in the stochastic Thiele equations w.r.t. G1 and G2.

Proposition 4.8. For each i ∈ 1, 2 and t > 0 we have

biJ+1(t) := bGi−

1(t) = E[bJ+1(t) | Git−],

βi(J+1)(J+2)(t) := βGi−

11,2(t, (t, J + 2)) = E[b(J+1)(J+2)(t) | Git−, δ = t]

almost surely on Zt− = J + 1,

biJ+2(t) := 1η<tbGi−

1,2(t) + 1η≥tbGi−

2(t) = E[bJ+2(t) | Git−]

almost surely on Zt− = J + 2,

Ri(J+1)(J+2)(t) :=βi(J+1)(J+2)(t) + Y Gi

J+2(t)− Y Gi

J+1(t)

=E[β(t) + Y (t) | Git−, δ = t]− E[Y (t) | Gi

t−, Zt = J + 1]

almost surely on Zt− = J + 1, and for each t ≥ 0 and j ∈ 1, . . . , J we have

L2j(J+1)(t) :=E[Y (t) | η = t, Zη− = j]− YG2

11(t)

=E[Y (t) | G2t , η = t, Zη− = j]− E[Y (t) | G2

t , Zt− = J + 1]

almost surely on Zt = J + 1.

25

Sketch of proof. Combine suitably the contents of Example 3.15, Proposition 3.7, the construc-tions of G1 and G2 according to the proof of Lemma 2.4, and (3.3).

4.3.2. Stochastic Thiele equations

Based on the characterization of relevant IF- and IB-compensators and state-wise quantities fromSubsection 4.3.1, the following two theorems yield stochastic Thiele equations for the classicstate-wise prospective reserves w.r.t. non-monotone information G1 and G2.

Theorem 4.9. The classic state-wise prospective reserves (Y G1

j )j∈S almost surely satisfy Y G1

j = Vjfor j ∈ 1, . . . , J and

0 = 1Zt−=J+1

(

Y G1

J+1(dt)− Y G1

J+1(t−)κ(dt)

κ(t−)+ b1J+1(t)µ(dt) +R1

J+1(J+2)(t) ΓG1−

(J+1)(J+2)(dt)

)

,

0 = 1Zt−=J+2

(

Y G1

J+2(dt)− Y G1

J+2(t−)κ(dt)

κ(t−)+ b1J+2(t)µ(dt)

)

.

Theorem 4.10. The classic state-wise prospective reserves (Y G2

j )j∈S almost surely satisfy Y G2

j =Vj for j ∈ 1, . . . , J and

0 = 1Zt−=J+1

(

Y G2

J+1(dt)− Y G2

J+1(t−)κ(dt)

κ(t−)+ b2J+1(t)µ(dt) +R2

J+1(J+2)(t) ΓG2−

(J+1)(J+2)(dt)

−J∑

k=1

L2k(J+1)(t) Γ

G2

k(J+1)(dt)

)

,

0 = 1Zt−=J+2

(

Y G2

J+2(dt)− Y G2

J+2(t−)κ(dt)

κ(t−)+ b2J+2(t)µ(dt)

)

.

Sketch of proof of Theorem 4.9 and Theorem 4.10. Since Zt = J + 1 = η ≤ t < δ = Zt =1, Zt = J + 2, η ≤ t = Zt = 1, 2, and Zt = J + 2, η > t = Zt = 2 for all t ≥ 0,following along the lines of the proof of Corollary 4.5 and pointing to Example 3.15 yields

1Zt−=J+1YGi

J+1(dt) = I1(t−)YGi

1(dt),

1Zt−=J+2YGi

J+2(dt) = I1,2(t−)YGi

1,2(dt) + I2(t−)YGi

2(dt)

almost surely. Now apply Theorem 4.2, calculate the terms explicitly, collect them, and applyProposition 4.7 and Proposition 4.8.

Remark 4.11. Note that the term

1Zt−=J+1

J∑

k=1

L2k(J+1)(t) Γ

G2

k(J+1)(dt)

can be replaced by

1Zt−=J+1L2•(J+1)(t) Γ

G2

•(J+1)(dt),

where

L2•(J+1)(t) := E[Y (t)|η = t]− Y G2

J+1(t),

ΓG2

•(J+1)(dt) :=1η≤t<δ

P (η ≤ t < δ)P (η ∈ dt).

To see this, apply Proposition 4.7, Proposition 4.8, and (3.4).

26

The stochastic differential equations that follow from Theorem 4.9 and Theorem 4.10 are fun-damentally different from the stochastic differential equations appearing in the presence of mono-tone information. Since Y Gi

j almost surely equals Vj for j ∈ 1, . . . , J, Corollary 4.5 yields thestochastic differential equations

0a.s.= 1Zt−=j

(

Y Gi

j (dt)− Y Gi

j (t−)κ(dt)

κ(t−)+ bj(t)µ(dt) +

∑

k:k 6=j

(

bjk(t) + Vk(t)− Y Gi

j

)

Λjk(dt)

)

for j ∈ 1, . . . , J. The sum at risks for k ∈ J + 1, J + 2 take an unusual form as they

involve VJ+1 and VJ+2 rather than Y Gi

J+1 and Y Gi

J+2. Since information discarding occurs upon orafter retirement and death, this just reflects full utilization of all available information (beforeretirement and death).

Another fundamental difference is evident in Theorem 4.10. Recall that G2 does not havethe time since retirement as admissible information. Referring to Remark 4.11, the stochasticdifferential equation for Y G2

J+1 includes the term

(

E[Y (t)|η = t]− Y G2

J+1(t)

)

1η≤t<δ

P (η ≤ t < δ)P (η ∈ dt).

It adjusts the dynamics to take into account the possibility that retirement has just occurred. Inthis case, at time t one would reserve E[Y (t)|η = t] rather than the averaged Y G2

J+1(t). This con-stitutes the first part of the product, while the second part is exactly the infinitesimal probabilityof retirement having just occurred, conditionally on the insured presently being retired.

4.3.3. Feynman-Kac formulas

We now specialize and simplify the setting to provide a more straightforward and less technicaldiscussion of the general results and their relation to actuarial practice.

Suppose that Z is semi-Markovian such that the F-compensators Λ of N admit densities w.r.t.the Lebesgue measure and such that (η, δ) is a continuous random variable. Denote by f(η,δ) thejoint density function of (η, δ), by fη|δ the conditional density function of η given δ, and by fη andfδ the marginal density functions of η and δ. Further, suppose that bj and bjk are deterministicfor all j, k ∈ S, j 6= k, and let there be a maximal contract time n < ∞, i.e. each bj and bjk isconstantly zero on the interval (n,∞).

Because of Proposition 2.3, the compensators Λ have representations of the form

Λjk(dt) = 1Zt−=jαjk(t, t− Ut−) dt, j ∈ 1, . . . , J, k ∈ 1, . . . , J + 2 \ j,

Λ(J+1)(J+2)(dt) = 1Zt−=J+1α(J+1)(J+2)(t, t− Uht−, t− U r

t−,Ht−) dt

for deterministic functions αjk and α(J+1)(J+2), so-called transition rates.The next results provide Feynman-Kac formulas that can serve as the starting point for the

development of numerical schemes for the classic state-wise prospective reserves (Vj)i∈S .

Proposition 4.12. Suppose the assumptions from the beginning of this subsection hold. If the

function WJ+2(·) is a bounded cadlag solution of

WJ+2(dt) =WJ+2(t−)κ(dt)

κ(t−)− bJ+2(t)µ(dt), t > 0, (4.5)

27

with terminal condition WJ+2(n) = 0, and the function WJ+1(·, ·, ·, ·) is a bounded and cadlag

solution of

WJ+1(dt, s, r, k) =WJ+1(t−, s, r, k)κ(dt)

κ(t−)− bJ+1(t)µ(dt)

−(

b(J+1)(J+2)(t) +WJ+2(t)−WJ+1(t, s, r, k))

α(J+1)(J+2)(t, s, r, k) dt,

t > r > s ≥ 0, k ∈ 1, . . . J,

(4.6)

with terminal conditions WJ+1(n, s, r, k) = 0 for 0 ≤ s ≤ r ≤ n and k ∈ 1, . . . , J, and the

functions Wj(·, ·), j ∈ 1, . . . , J, are bounded and cadlag solutions of

Wj(dt, s) =Wj(t−, s)κ(dt)

κ(t−)− bj(t)µ(dt)+

−∑

k≤J :k 6=j

(

bjk(t) +Wk(t, t)−Wj(t, s))

αjk(t, s) dt

−(

bj(J+1)(t) +WJ+1(t, s, t, j) −Wj(t, s))

αj(J+1)(t, s) dt

−(

bj(J+2)(t) +WJ+2(t)−Wj(t, s))

αj(J+2)(t, s) dt,

t > s ≥ 0, j ∈ 1, . . . J,

(4.7)

with terminal conditions Wj(n, s) = 0 for 0 ≤ s ≤ n, then for all t ≥ 0 and j ∈ 1, . . . , J,

1Zt=jWj(t, t− Ut) = 1Zt=j Vj(t) = 1Zt=jV (t)

almost surely, and for all t ≥ 0,

1Zt=J+1WJ+1(t, t− Uht , t− U r

t ,Ht) = 1Zt=J+1VJ+1(t) = 1Zt=J+1V (t),

1Zt=J+2WJ+2(t) = 1Zt=J+2 VJ+2(t) = 1Zt=J+2V (t)

almost surely.

Proof. Note that the right-continuity of the solutions of the differential/integral equations allowsus to uniquely expand the domains of the solutions to t ≥ s ≥ 0, t ≥ r > s ≥ 0 and t ≥ 0. Thatmeans that Wj(t, t) and WJ+1(t, s, t, k) are indeed given by the solutions.

Since the bounded and cadlag solution WJ+2 of (4.5) is deterministic, it is also F-predictableand by multiplying (4.5) with 1Zt−=J+2 we obtain (4.4) for j = J + 2. By multiplying equa-

tion (4.6) with 1Zt−=J+1 and replacing s, r and k by t − Uht−, t − U r

t−, and Ht−, we obtain

that WJ+1(t, t−Uht−, t−U r

t−,Ht−) is an F-predictable, bounded, and cadlag solution of (4.4) forj = J + 1.

Multiplying equation (4.7) with 1Zt−=j1τi<t≤τi+1 and replacing s by τi1Zτi=j + t1Zτi 6=j,we obtain that Wj(t, τi1Zτi

=j + t1Zτi6=j), j ∈ 1, . . . , J, is a solution of (4.4) on the interval

(τi, τi+1]. This follows from the almost sure identities

1Zt−=jWk(t, τi1Zτi=k + t1Zτi 6=k) = 1Zt−=jWk(t, t),

1Zt−=jWJ+1(t, τi1Zτi=j + t1Zτi 6=j, 0, j) = 1Zt−=jWJ+1(t, t− Uht−, U

rt−,Ht−)

for all t ∈ (τi, τi+1] and j, k ∈ 1, . . . , J, j 6= k. Summing over i ∈ N0 yields that the boundedand cadlag F-predictable processes

Wj(t, t− Ut−1Zt−=j) =

∞∑

i=0

1τi<t≤τi+1Wj(t, τi1Zτi=j + t1Zτi 6=j), j ∈ 1, . . . , J,

28

are solutions of (4.4) for j ∈ 1, . . . , J due to the fact that

1Zt−=jWk(t, t− Ut−1Zt−=k) = 1Zt−=jWk(t, t)

almost surely for all t > 0 and j, k ∈ 1, . . . , J, j 6= k.All in all, we conclude that the processes Wj(t, t − Ut−1Zt−=j), j ∈ 1, . . . , J, WJ+1(t, t −

Uht−, t−U

rt−,Ht−), andWJ+2(t) form an F-predictable bounded and cadlag solution of the equation

system (4.4), which implies that, according to Proposition 4.6, they equal the classic state-wiseprospective reserves Vj(t) on Zt = j for j ∈ 1, . . . , J + 2. Since Zt = J + 1 implies η ≤ t,we may replace 1Zt=J+1WJ+1(t, t−Uh

t−, t−U rt−,Ht−) by 1Zt=J+1WJ+1(t, t−Uh

t , t−U rt ,Ht).

Moreover, we have

1Zt=jVj(t, t− Ut) = 1Zt=jWj(t, t− Ut−1Zt−=j), j ∈ 1, . . . , J,

almost surely for all t ≥ 0 under the conventions U0− := 0 and Z0− := Z0. This implies thestatement of the Proposition.

The numerical schemes that can be developed based on Proposition 4.12 are significantly morecomplex than in the classic (semi-)Markovian case, see e.g. Adekambi and Christiansen (2017).The sum at risks involve WJ+1(t, s, t, j), which must be computed based on (4.6) for all 0 ≤ s < tusing e.g. the method of lines.

Recall that Y Gi

j = Vj almost surely for j ∈ 1, . . . , J, cf. Theorem 4.9 and Theorem 4.10,

and due to the assumptions given at the beginning of this subsection, we also have Y Gi

J+2 = VJ+2

almost surely. The next results provide Feynman-Kac formulas for the residuary classic state-wiseprospective reserve in the presence of non-monotone information G1 and G2. Proofs are given atthe end of the subsection.

Proposition 4.13. Suppose the assumptions from the beginning of this subsection hold. If

W 1J+1(·, ·) is a bounded and cadlag solution of

W 1J+1(dt, r) =W

1J+1(t−, r)

κ(dt)

κ(t−)− bJ+1(t)µ(dt)

−(

b(J+1)(J+2)(t) +WJ+2(t)−W 1J+1(t, r)

)

α1(J+1)(J+2)(t, r) dt

(4.8)

for 0 < r < t with terminal conditions W 1J+1(n, r) = 0 for 0 ≤ r ≤ n, and where WJ+2(·)

solves (4.5) while

α1(J+1)(J+2)(t, r) :=

fδ|η(t|r)

P (δ ≥ t | η = r), 0 ≤ r ≤ t, (4.9)

then 1Zt=J+1W1J+1(t, η) = 1Zt=J+1 Y

G1

J+1(t) = 1Zt=J+1YG1

(t) almost surely for all t ≥ 0.

Proposition 4.14. Suppose the assumptions from the beginning of this subsection hold. If

W 2J+1(·, ·) is a bounded and cadlag solution of

W 2J+1(dt) =W

2J+1(t−)

κ(dt)

κ(t−)− bJ+1(t)µ(dt)

−(

b(J+1)(J+2)(t) +WJ+2(t)−W 2J+1(t)

)

α2(J+1)(J+2)(t) dt

+(

W 1J+1(t, t)−W 2

J+1(t))

ξJ+1(t) dt

(4.10)

29

for 0 < t with terminal condition W 2J+1(n) = 0, and where WJ+2(·) and W 1

J+1(·, ·) solve (4.5)and (4.8) while

α2(J+1)(J+2)(t) :=

∫ t0 f(η,δ)(s, t) ds

P (η < t ≤ δ), (4.11)

ξJ+1(t) :=fη(t)

P (η ≤ t < δ), (4.12)

then 1Zt=J+1W2J+1(t) = 1Zt=J+1Y

G2

J+1(t) = 1Zt=J+1YG2

(t) almost surely for all t ≥ 0.

In order to reduce the computation time and simplify actuarial modeling and statistical es-timation, practitioners, when computing the prospective reserve for non-retirees based on Wj ,j ∈ 1, . . . , J, often approximate WJ+1(t, s, t, j) by a less complex quantity such as W 1

J+1(t, t),which discards information concerning previous health records, or W 2

J+2(t), which additionallydiscards information concerning the time of retirement. Replacing WJ+1 by W i

J+1 produces ap-proximation errors on the individual level (and redistribution of wealth on the portfolio level fornon-retirees).

Proposition 4.13 and Proposition 4.14 can be used to develop computational schemes for W 1J+1

and W 2J+1, respectively. Focusing on W 2

J+1, this involves the transition rate α2(J+1)(J+2), which

by (4.11) is the hazard rate corresponding to a classic mortality table for retirees. It also involvesthe adjustment term

(

W 1J+1(t, t)−W 2

J+1(t))

ξJ+1(t) dt,

where according to (4.12), ξJ+1(t) dt is the infinitesimal probability of retirement having justoccurred (at time t), conditionally on the insured presently being retired.

If the mortality does not depend on the time since retirement, i.e. if α1(J+1)(J+2)(t, r) ≡

α2(J+1)(J+2)(t), we end up with the differential/integral equations

W iJ+1(dt) =W i

J+1(t−)κ(dt)

κ(t−)− bJ+1(t)µ(dt)

−(

b(J+1)(J+2)(t) +WJ+2(t)−W iJ+1(t)

)

α2(J+1)(J+2)(t) dt.

(4.13)

Even though the mortality of retirees might depend on the time since retirement, practitionersoften still utilize (4.13) directly. This produces additional approximation errors on the individuallevel (and redistribution of wealth on the portfolio level for retirees as well as non-retirees).

Proof of Proposition 4.13. Note that (4.8) implies that W 1J+1(·, η) has paths of finite variation on

compacts. By applying integration by parts, we obtain

1η<t d(

1Zt=J+1v(t)W1J+1(t, η)

)

= 1Zt−=J+1

(

v(t)W 1J+1(dt, η)− v(t)WJ+1(t−, η)

κ(dt)

κ(t−)

)

− v(t)W 1J+1(t, η)N(J+1)(J+2)(dt).

almost surely. Inserting (4.8) into the latter term leads to

1η<t d(

1Zt=J+1v(t)W1J+1(t, η)

)

= −1Zt−=J+1v(t)B(dt)− v(t)WJ+2(t)N(J+1)(J+2)(dt)

+ v(t)R1(J+1)(J+2)(t)M

1(J+1)(J+2)(dt)

30

almost surely; here R1(J+1)(J+2)(t) := b(J+1)(J+2)(t)+WJ+2(t)−W

1J+1(t, η) andM

1(J+1)(J+2)(dt) :=

N(J+1)(J+2)(dt)−1Zt−=J+1α1(J+1)(J+2)(t, η) dt. Thus, since η < t ⊆ η < s for s ≥ t ≥ 0, we

find that almost surely for all t ≥ 0,

1η<t1Zt=J+1v(t)W1J+1(t, η)

= E[1η<t1Zt=J+1v(t)W1J+1(t, η) | G

1t ]

= 1η<tE

[

v(t)

∫

(t,n]1Zs−=J+1

κ(t)

κ(s)B(ds)

∣

∣

∣

∣

G1t

]

+ 1η<tE

[∫

(t,n]v(s)WJ+2(s)N(J+1)(J+2)(ds)

∣

∣

∣

∣

G1t

]

− 1η<tE

[∫

(t,n]v(s)R1

(J+1)(J+2)(s)M1(J+1)(J+2)(ds)

∣

∣

∣

∣

G1t

]

= 1η<t1Zt=J+1v(t)YG1

(t)

− 1η<t1Zt=J+1E

[∫

(t,n]v(s)R1

(J+1)(J+2)(s)M1(J+1)(J+2)(ds)

∣

∣

∣

∣

G1t

]

Recall that Zt = J + 1 = Z ∈ 1. Pointing to Proposition 3.7, the constructions of G1

according to the proof of Lemma 2.4, and (3.3), straightforward calculations then yield that thelast line equals zero. All in all, we conclude that

1η<t1Zt=J+1v(t)W1J+1(t, η) = 1η<t1Zt=J+1v(t)Y

G1

(t) = 1η<t1Zt=J+1v(t)YG1

J+1(t)

almost surely for all t ≥ 0. Since v and Y G1

almost surely have cadlag sample paths, cf. Proposi-tion 3.8, we may replace 1η<t1Zt=J+1 by 1η≤t1Zt=J+1 = 1Zt=J+1. Using v > 0 completesthe proof.

Proof of Proposition 4.14. Note (4.10) implies that W 2J+1(·) has paths of finite variation on com-

pacts. By applying integration by parts, inserting (4.10), applying Theorem 4.10, and referringto Remark 4.11, straightforward calculations yield

d(

1Zt=J+1v(t)W2J+1(t)− 1Zt=J+1v(t)Y

G2

J+1(t))2

= v(t)21Zt−=J+1

(

W 2J+1(t)− Y G2

J+1(t))2(

α2(J+1)(J+2)(t)− ξJ+1(t)

)

dt

− v(t)2(

W 2J+1(t)− Y G2

J+1(t))2(N(J+1)(J+2)(dt)− 1Zt−=J+1α

2(J+1)(J+2)(t) dt

)

+ v(t)2(

W 2J+1(t)− Y G2

J+1(t))2(

− 1Zt=J+1ξJ+1(t) dt+

J∑

k=1

Nk(J+1)(dt)

)

almost surely. Following along the lines of the proof of Proposition 4.13, we find that

v(t)2P (Zt = J + 1)(

W 2J+1(t)− Y G2

J+1(t))2

= E[

1Zt=J+1v(t)2(

W 2J+1(t)− Y G2

J+1(t))2]

= −E

[∫ n

tv(s)21Zs−=J+1

(

W 2J+1(t)− Y G2

J+1(t))2(α2(J+1)(J+2)(s)− ξJ+1(s)

)

ds

]

= −

∫ n

tv(s)2P (Zs = J + 1)

(

W 2J+1(t)− Y G2

J+1(t))2(α2(J+1)(J+2)(s)− ξJ+1(s)

)

ds

31

almost surely. This means that the function

f(t) := v(t)2P (Zt = J + 1)(

W 2J+1(t)− Y G2

J+1(t))2

almost surely satisfies the integral equation

f(t) = −

∫ n

tf(s)

(

α2(J+1)(J+2)(s)− ξJ+1(s)

)

ds

for all t ∈ [0, n] under the convention (n, n] = ∅. Note that

|f(t)| ≤

∫ n

t|f(s)|

∣

∣α2(J+1)(J+2)(s)− ξJ+1(s)

∣

∣ ds

almost surely for all t ∈ [0, n]. According to the the backward Gronwall inequality (see Lemma 4.7in Cohen and Elliott, 2012), f(t) = 0 almost surely for all t ∈ [0, n]. Since v > 0, for each t ≥ 0

it then holds that 1Zt=J+1W2J+1(t) = 1Zt=J+1 Y

G2

J+1(t) almost surely. Since the implicatedprocesses almost surely have cadlag sample paths, cf. Corollary 3.9, there exists a joint P -nullset. Thus 1Zt=J+1W

2J+1(t) = 1Zt=J+1 Y

G2

J+1(t) almost surely for all t ≥ 0 as desired.

Acknowledgments and declarations of interest

Christian Furrer’s research is partly funded by the Innovation Fund Denmark (IFD) under FileNo. 7038-00007. The authors declare no competing interests.

A. Proofs

Proof of Proposition 2.3. As a consequence of Assumption 2.2, the only non-trivial statement ofthe proposition relates to intertemporal dependency structure of Z after retirement, so it sufficesto study the quantities

P (Zs = J + 1 | Ft )

on the event Zt = J+1 for 0 ≤ t < s <∞ . To this end, consider sets Ant := Zt = J+1, N (t) =

n, n ∈ N, where N = (N(t))t≥0 is the process counting the total number of jumps of Z given by

N(t) =∑

j,k∈Sj 6=k

Njk(t), t ≥ 0,

and denote with τ = (τi)i∈N and τ = (τi)i∈N the point processes corresponding to the jump timesof Z and Z, respectively. Fix 0 ≤ t < s < ∞, and fix n ∈ N. On An

t it then almost surely holdsthat

τn = τn = η, Zτn ∈ J + 1, . . . , 2J,

τi = τi Zτi = Zτi ∈ 1, . . . , J, ∀i = 1, . . . , n− 1.

In particular,

P (Zs = J + 1 | Ft )1Ant

a.s.= P

(

Zs ∈ J + 1, . . . , 2J∣

∣

∣τn, Zτn , τn−1, Zτn−1

, . . . , τ1, Zτ1

)

1Ant.

32

Suppose (Z, U ) is Markovian such that Z is semi-Markovian. By the law of iterated expectationsand the strong Markov property, cf. Theorem 7.5.1 in Jacobsen (2006), it follows that

P(

Zs ∈ J + 1, . . . , 2J∣

∣

∣τn, Zτn , τn−1, Zτn−1

, . . . , τ1, Zτ1

)

1Ant

= E[

P(

Zs ∈ J + 1, . . . , 2J∣

∣

∣τn, Zτn , τn − τn−1

)∣

∣

∣τn, Zτn , τn−1, Zτn−1

, . . . , τ1, Zτ1

]

1Aj

t

a.s.= E

[

P(

Zs ∈ J + 1, . . . , 2J∣

∣

∣τn, Zτn , τn − τn−1

)∣

∣

∣τn, Zτn , τn−1, Zτn−1

]

1Aj

t

= P(

Zs ∈ J + 1, . . . , 2J∣

∣

∣τn, Zτn , τn−1, Zτn−1

)

1Ant.

Thus on Ant = Zt = J + 1, N(t) = n it almost surely holds that

P (Zs = J + 1 | Ft ) = P(

Zs = J + 1∣

∣

∣τn, Zτn , τn−1, Zτn−1

)

= P(

Zs = J + 1∣

∣

∣t− τn, Zτn , t− τn−1, Zτn−1

)

= P(

Zs = J + 1∣

∣

∣U rt , Zt, U

ht ,Ht

)

,

which does not depend on n. We conclude that if (Z, U ) is Markovian, then on Zt = J + 1,

P (Zs = J + 1 | Ft )a.s.= P

(

Zs = J + 1∣

∣

∣U rt , Zt, U

ht ,Ht

)

proving the first part of the proposition. The proof of the second and final part follows by similararguments.

Proof of Lemma 2.4. Let N− = (N−(t))t≥0 be the process counting the number of jumps of Zexcept retirement and death given by

N−(t) =∑

j,k∈Sk/∈j,J+1,J+2

Njk(t), t ≥ 0,

and denote by (τ−i )i∈N the point process corresponding to the jumps of N−. The σ-algebras (2.1)and (2.2) are equivalent to G1

t and G1t−, respectively, if we set

T1 = η, S1 = ∞, ζ1 = (T1, ZT1),

T2 = δ, S2 = ∞, ζ2 = (T2, ZT2),

T2+i = τ−i , S2+i = T1 ∧ T2, ζ2+i = (T2+i, ZT2+i), i ∈ N.

If we replace ζ1 = (T1, ZT1) by the constant ζ1 = (0, Zτ1) = (0, J + 1), then (2.1) and (2.2) are

equivalent to G2t and G2

t−, respectively.

Proof of Proposition 3.5. Since Y is integrable, for each j ∈ S and t ≥ 0 the mapping

Ct,j ∋ A 7→ νt,j(A) :=

∫

AY (t) dmt,j

is a finite signed measure on Ct,j which is absolutely continuous with respect to the sub-probabilitymeasure mt,j given by

Ct,j ∋ A 7→ mt,j(A) = P (A ∩ Zt = j).

33

According to the Radon-Nikodym theorem there exist mappings ω 7→ Yj(t)(ω) that are Ct,j-measurable and satisfy

νt,j(A) =

∫

AYj(t) dmt,j, A ∈ Ct,j . (A.1)

In particular

∫

A∩Zt=jY (t) dP =

∫

A∩Zt=jYj(t) dP, j ∈ S,A ∈ Ct,j,

which by Lemma 3.1 yields

∫

AY (t)1Zt=j dP =

∫

AYj(t)1Zt=j dP, j ∈ S,A ∈ Ct.

We conclude that 1Zt=jYj(t)a.s.= 1Zt=jY (t) for each j ∈ S and t ≥ 0. This establishes existence

of the state-wise counterparts. Furthermore, if there is another real-valued random variable Yj(t)that has the properties of Yj(t), we necessarily have

0 =

∫

A∩Zt=j(Yj(t)− Yj(t)) dP =

∫

A×j(Yj(t)(ω)− Yj(t)(ω)) dmt(ω, j),

for A ∈ Ct,j, which means that the mapping (ω, j) 7→ Yj(t)(ω)− Yj(t)(ω) is mt-almost everywherezero. This establishes the desired uniqueness of the state-wise counterparts.

Proof of Lemma 3.6. If P (Zt = j) = 0, the result is trivial. Thus suppose P (Zt = j) > 0. SinceEt,j[E[X | Ct] | Ct,j ] is the conditional expectation of E[X | Ct] given Ct,j w.r.t. Pt,j , we find that

∫

AEt,j[E[X | Ct] | Ct,j ] dPt,j =

∫

AE[X | Ct] dPt,j .

Note that by definition of Ct,j, we have A ∩ Zt = j ∈ Ct. It follows that

∫

AEt,j [E[X | Ct] | Ct,j ] dPt,j =

1

P (Zt = j)

∫

A∩Zt=jE[X | Ct,j ] dP

=1

P (Zt = j)

∫

AX1Zt=j dP

=

∫

AX dPt,j,

where we have used that E[X | Ct] is the conditional expectation of X given Ct w.r.t. P . Inconclusion, Et,j[E[X | Ct,j ] | Ct,j ] is a version of the conditional expectation of X given Ct,j w.r.t.Pt,j which completes the proof.

Proof of Corollary 3.9. From Proposition 3.8 we know that Y G almost surely has cadlag pathsof finite variation on compacts. Since 1Zt=jY

Gj (t)

a.s.= 1Zt=jY

G(t), the path properties of Y G

also hold almost surely for t 7→ 1Zt=jYGj (t). Proposition 3.8 directly states the path properties

for t 7→ Ix(t)YGx (t). Since Assumption 2.1 and (2.3) imply that the paths of Z and Z have at

most a finite number of jumps on [0, t], the cadlag and finite variation property are still true fort 7→ 1Zt−=jY

Gj (t) and t 7→ Ix(t−)YG

x (t).

34

Proof of Proposition 3.11. Suppose that (Ti, Si, ζi)i∈N := (τi,∞, (Zτi , τi))i∈N, such that F = G.Fix t > 0 and j ∈ S. By (3.6) we almost surely find

∑

k∈Sj 6=k

YF−

kj (t) = 1Zt− 6=j

∞∑

n=0

1τn<t≤τn+1E[Y (t) | (Zτ1 , τ1), . . . , (Zτn , τn), (Zτn+1, τn+1) = (j, t)]

= 1Zt− 6=j

∞∑

n=0

1τn<t≤τn+1

E[1(Zτn+1,τn+1)=(j,t)Y (t) | (Zτ1 , τ1), . . . , (Zτn , τn)]

E[1(Zτn+1,τn+1)=(j,t) | (Zτ1 , τ1), . . . , (Zτn , τn)]

.

Since Zt− 6= j, Zt = j, τn < t ≤ τn+1 = Zt− 6= j, τn < t ≤ τn+1, τn+1 = t, Zτn+1= j for any

n ∈ N0, we further conclude on the basis of Example 3.2 and (3.2) that

∑

k∈Sj 6=k

YF−

kj (t) = 1Zt− 6=j

∞∑

n=0

1τn<t≤τn+1

E[1Zt=jY (t) | (Zτ1 , τ1), . . . , (Zτn , τn)]

E[1Zt=j | (Zτ1 , τ1), . . . , (Zτn , τn)]

= 1Zt− 6=j

∞∑

n=0

1τn<t≤τn+1E[Y (t) | Ft−, Zt = j]

= 1Zt− 6=jYFj (t)

almost surely. Similarly, YF−

jj (t)a.s.= 1Zt−=jY

Fj (t). Writing

Y Fj (t) = Y F

j (t)1Zt−=j + Y Fj (t)1Zt− 6=j

and collecting terms completes the proof.

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